ON POTENTIALS GENERATED BY INCREASING PROCESSES

ON POTENTIALS GENERATED BY INCREASING PROCESSES

VALENTIN GRECEA

In the frame of a filtered probability space, we consider some processes with finite variation which “approximate” the processes with bounded mean oscillation. Also, in the frame of a Ray process, we consider increasing processes which generate the supermartingales of class (D).

AMS 2000 Subject Classification: 60Gxx, 60Hxx, 60Jxx.

Key words: increasing process, supermartingale, Ray process.

1. INTRODUCTION

The importance of increasing processes in probability is well known. The purpose of this paper is to add some complements to the applications of in- creasing processes and potentials (resp. the left potentials) that they generate.

First, we consider the case of left potentials in the general frame of a filtered probability space, in the study of the space BMO of martingales with bounded mean oscillation, which occurs essentially in establishing the equivalence be- tween the maximal and the quadratic norms on martingales (the inequality of Davis, from which one derives the inequality of Burkholder onH^p forp >1).

Second, we consider the frame of a Markovian Ray process on a compact metric space and give a result concerning to the representation of (positive) supermartingales of class (D) in connection with the well known representation of potentials (excessive functions) by additive functionals. We assume that the reader is familiar with general martingale theory and general theory of Markov processes, as exposed in [1].

1. Let (Ω,Ft,F, P) be a filtered probability space such that the filtration (Ft) satisfies the usual conditions. We recall that the potential (resp. – left potential) generated by an increasing and right continuous, predictable and null at 0 (resp. optional, positive but not necessarilly null at 0) process (A_t) on [0,∞], is the process

Z_t=E[A_∞| Ft]−A_t (resp. Z_t^g =E[A_∞| Ft]−A_t−),

MATH. REPORTS9(59),3 (2007), 267–274

whereE[A_∞ | Ft] is a r.c.l.l. version of the martingale on [0,∞] (continuous at ∞!) taking the valueA_∞ at ∞, which dominates the process (A_t).

Of course, the supermartigales (Zt) and (Z_t^g) of class (D) (the second is not right continuous but is still optional) are properly potentials iff (A_t) does not jump at∞, and each of them determines the corresponding (At).

We define the (vector) spaces BMOp for p > 0 as follows: a r.c.l.l. on [0,∞] and (Ft) adapted real process (X_t) belongs to BMO_p iff there exists some constanta >0 such that

E[|X_S−X_T−|^P | FT]≤a^P

for any stopping timesT, S such thatT ≤S. One can see that considering the smallest constant as above, one defines a norm

BMOp on (the set of classes up to evanescence of)BMO_p, and it is known (see [1, VI, (109.7)]) that the spacesBMO_p coincide forp≥1 in the sense that the corresponding norms are equivalent. We writeBMO instead ofBMO₁.

Theorem 1.1. Let (X_t)_0≤t≤∞ be a real r.c.l.l. and (Ft) adapted pro- cess. Then (X_t) belongs to BMO iff it admits a decomposition X_t=U_t+K_t, where (Ut) is a process (r.c.l.l. on [0,∞] and (Ft)-adapted) with finite vari- ation such that its variation generates a bounded left potential, and (K_t) is a bounded process. Moreover, if a denotes the BMO₁ norm of (X_t), one can find a decomposition of Xt as above such that the left potential generated by the variation of U_t is bounded by 6a, and |K_t| ≤2a on [0,∞].

Proof. The implication “⇐” is almost obvious by using the relations (S, T stopping times,S ≥T)

|U_S−U_T−| ≤

[T,S]|dU_s| ≤

[T,∞]|dU_s|.

For the converse, we may assume that X_∞ = X_∞− by removing the bounded jump of (Xt) at ∞, which will be assigned to (Ut) or (Kt), as we wish, and we work in the rest of the proof with processes on [0,∞). Denote byathe BMO₁ norm of (X_t) and fix a constantc > a.

Let the sequence (T_n) of stopping times defined as (1.1) T₀ = 0, T_n+1 = inf{t > T_n:|X_t−X_Tn|> c} (of course we put inf∅=∞), the increasing process

(1.2) A_t=

n≥0

I_{Tn≤t}, A_∞= lim

t→∞A_t

and, finally,

(1.3) U_t=X₀+

n≥1

(X_Tn−X_Tn−1)I_{Tn≤t}, U_∞= lim

t→∞U_t. We show first that, with A₀₋= 0 by definition,

(1.4) E[A_∞−A_T−| FT]≤c/(c−a)

for any stopping time T (which means that (A_t) is an integrable increasing process with bounded left potential). Hence it will then follow that

(1.5) E

[T,∞]|dUs| | FT

≤c(c+a)/(c−a)

for any stopping timeT because of the relation

(1.6) |dU_s(ω)| ≤(c+a)dA_s(ω), ω∈Ω, obviously implied by the inequality

(1.7) |X_Tn−X_Tn+1| ≤c+a, n∈N^∗,

consequence of the construction of (Tn) and the fact that |∆Xt| ≤ a by hy- pothesis (|X₀| ≤a, too!).

So, returning to show (1.4), we fix a stopping timeT, remark thatTn→

∞ since (X_t) has (finite) left limits by hypothesis, and consider the random variable nT = inf{n : T ≤ Tn}. If we put Sn = Tn_T+n(T_∞ = ∞), one can easily see that (Sn) is an increasing sequence of stopping times. From the construction of (T_n) we deduce

(1.8) cI_{Sn+1<∞}≤ |X_Sn+1−X_Sn|(≤c+a), n∈N^∗, and since the hypothesis implies the relation

(1.9) E[|XV −XU| | FU]≤a

for any stopping times U, V such that U ≤ V (we apply the hypothesis to U+_n¹,V +¹_n and letn→ ∞) we can write

(1.10) cE[Sn+1 <∞ | FS_n]≤E[|XS_n+1−XS_n| | FS_n]≤aI_{Sn<∞}

for any n∈N^∗. By inverse recurrence we get

(1.11) E[S_n+1 <∞ | FS₀]≤(a/c)ⁿ, n∈N^∗. We remark thatA_∞−A_T−=

n≥0I_{Sn<∞}, hence (1.12) E[A_∞−AT−| FT] =

n≥−

E[Sn<∞ | FT]≤c/(c−a).

Finally, it follows by the construction of (U_t) that |X_t−U_t|is a process bounded byc, which completes the proof by taking c= 2a.

Corollary([1, VI, 109]). The norms

BMOp on BMO are all equiv- alent forp≥1.

New proof. Since clearly

BMOp ≥ _BMO

q for p > q by Jensen’s inequality, it suffices to show that for anyn ∈ N there exists some constant Cn>0 such that

BMOn ≤Cn BMO1. Let (X_t) ∈ BMO, denote a = X_BMO

1 and consider a decomposition X_t = K_t+U_t such that |K_t| ≤ 2a and E[

[T,∞]|dU_s| | FT] ≤ 6a for any stopping timeT. Denote for simplicityBt=

[0,t]|dUs|, which is an increasing process with bounded left potential, and fix T as above. For any n ∈ N we have

(1.13)

E

sups≥T|X_s−X_T−|

n

| FT

≤

≤E

sups≥T|Ks−KT−|+

[T,∞]|dUs|

n

| FT

≤

≤E _n

k=0

C_n^k(4a)^k(B_∞−BT−)^n−k | FT

.

But it is well known and “classical” (see [1, VI, 106 c)]) that relatively to theincreasingprocess (Bt) we have

(1.14) E[(B_∞−B_T−)^m | FT]≤m!(6a)^m, m∈N.

Returning to (1.13), we see that there exists some constant C_n (for eachn) such that

(1.15) E

sups≥T|X_s−X_T−|n

| FT

≤C_naⁿ,

which implies of course the desired conclusion. In addition, for n = 1, the above relation implies that the increasing process A_t= sup_s≤t|X_s| generates a bounded left potential.

Remarks. We recall that the subspace of BMOp consisting of uniformly integrable martingales continuous at ∞ is known as BMO_p, and BMO₁ is isomorphic to the dual of the Banach spaceH_∗¹ (with maximal norm), subject to a theory essentially due to Herz and Lepingle, developped in [1, VII, 70–80].

It follows from this theory that the set {X_∞ : (Xt) ∈ BMO} coincides with the set{A_∞: (A_t) is a process with finite variation on [0,∞] (r.c.l.l. and (Ft) adapted) whose variation generates a bounded left potential}, which coincides according to our result to the set{X_∞: (Xt)∈BMO}.

We note that the martingale property of elements of BMO occurs “dis- cretely” in establishing the duality between H_∗¹ and BMO by above quoted

“maximal” theory, where the set{X_∞: (Xt)∈BMO}is of interest. But when establishing the equivalence between maximal and quadratic norms onH¹(the inequality of Davis) one considers BMO₂ (which is isomorphic to BMO₁) as imbeded in the space H² of square integrable martingales, where the square bracket [,] is defined and the equations below hold (for any (Xt)∈H²):

(1.16) E[(X_∞−X_T−)² | FT] =E[[X, X]_∞−[X, X]_T− | FT] for any stopping timeT, and

(1.17) E[M_∞X_∞] =E[[M, X]_∞]

for any (Mt)∈H². In fact, the second equation follows from the first by taking T = 0 and polarization. The reader could check the role of above relations for (Xt)∈BMO in different proofs of the inequality of Davis (see for example [1, VII, 90]). The right hand of 1.16 suggests to consider the “quadratic” BMO norm. See, for example, [3] for an extension of the “quadratic” theory.

2. On a compact metric space F we let (Pt)t≥0 be a Ray semigroup which is supposed to be Markovian (P_t1 = 1). We consider the canonical Ray Markovian process (Ω,Ft,F,Θt, Xt, P^x) onF with transition semigroup (P_t). We refer the reader for definition and theory of Ray semigroups and Ray processes to [1, XIV] and [2]. Here, we recall that P₀ = Id in general (P₀ = Id characterizes the Feller semigroups) and if N denotes the set of nonbranch points ofF, then the “path space” Ω is the set of right continuous mappings ω : [0,∞) → N such that ω(t−) exists in F for any t > 0; we putX_t(ω) = ω(t) for t ≥ 0. In most situations, there exists a distinguished absorbing point (relatively to (Pt)) ∆∈F called “death point” or “cemetery”.

Clearly, ∆∈N, and in this case the subset of Ω consisting of paths with “life time” carries all probabilities P_x^x_∈N and can be also considered a path space.

Theorem 2.1. Let (Y_t) be a positive and finite r.c.l.l. and (Ft)-adapted process on Ω such that (Y_t) is a supermartingale of class (D) with respect to P^x for any x ∈ N. Then there exists an increasing r.c. and (Ft)-adapted process (A_t), predictable and null at0, unique up to evanescence such that for any x∈N it is integrable with respect to P^x and we have

(2.1) Y_t=E^x[A_∞−A_t| Ft] +E^x[Y_∞| Ft] P^x-a.s., where Y_∞= lim inft→∞Yt.

Proof. We first note thatY_∞is integrable with respect to anyP^xbecause (Y_t) is a positive supermartingale and, moreover, Y_∞ = lim_t→∞Y_t P^x-a.s.

(see [1, VI, 6]).

Fix x ∈ N. The process Y_t =Yt−E^x[Y_∞ | Ft^x], where (E^x[Y_∞ | Ft^x]) is considered as a r.c.l.l. martingale dominated by (Y_t), is a potential of class (D) with respect to P^x. It is then well known (see [1, VII, 8]) that there exists a unique P^x integrable increasing predictable process (B_t^x) null at 0, r.c., (F_t^x)-adapted, such that

(2.2) Yt=E^x[B_∞^x −B_t^x| Ft^x] +E^x[Y_∞| Ft^x] P^x-a.s.

We look to find a common version of all processes (B_t^x) over x ∈ N, which should be increasing, null at 0, r.c., (Ft) adapted. For this purpose we first use the standard approach (see [1, VII, 22]) by “close Laplacians”

(2.3) ∆ⁿ_t =nE^x[Y_t−Y_t+1

n | Ft^x] =nE^x[Y_t−Y_t+1 n | Ft^x].

In fact, (∆ⁿ_t) is the difference between the supermartingale (nY_t) and a r.c.l.l.

version of the supermartingale (nE^x[Y_t+1

n | Ft^x]) dominated by (nYt), so that (∆ⁿ_t) is a positive, r.c.l.l. and (Ft^x) adapted process. As in [1, VII, 22], we take

(2.4) B_tⁿ=

[0,t]∆ⁿ_sds

which converges to B^x_t in the weak topology σ(L¹, L^∞) with respect to P^x. Using now a slight modification of the selection lemma [1, XV, 2b)] applied to B_t^x(x∈N) for fixed t, we can find a (Ft)-measurable random variableBtsuch thatB_t=B_t^xP^x-a.s. for anyx∈N, if we are able to show that the mapping

x→E^x[ϕBⁿ_t]

is universally measurable for any (fixed) boundedFt^◦-measurable function ϕ and for anyn∈N. We have

(2.5)

E^x[Bⁿ_tϕ] =E^x

[0,t]∆ⁿ_sds

ϕ

=

[0,t]E^x[∆ⁿ_sϕ]ds n

[0,t]E^x[E^x[Y_s−Y_s+1

n | Fs^◦]ϕ]ds=n

[0,t]E^x[(Y_s−Y_s+1

n)E^x[ϕ|Fs^◦]]ds.

Using the monotone class theorem, in order to check the (universal) measurability inxof the last expression, we may consider just the case where ϕ=f₁◦Xt₁f₂◦Xt₂· · ·fn◦Xt_n, 0 ≤t₁ < t₂· · ·< tn =t, with fi Borel and bounded onN fori= 1,2, . . . , n. Using now the Markov property, we have (2.6)

[0,t]E^x[(Y_s−Y_s+1

n)E^x[ϕ| Fs^◦]]ds=

=

n−1 k=0

[t_k,t_k+1)E^x[(Y_s−Y_s+1

n)E^x[ϕ| Fs^◦]]ds=

=

n−1 k=0

[t_k,t_k+1)E^x[(Ys−Y_s+1

n)f₁◦Xt₁. . . fk◦Xt_kPt_k+1−sgk(Xs)]ds, whereg_n−1 =f_n, and fork= 0,1, . . . , n−2 we put

g_k(·) =f_k+1(·)P_tk+2−tk+1(·, dx_k+2). . . P_tn−tn−1(x_n−1, dx_n)f_k+2(x_k+2). . . f_n(x_n).

Of course, allg_kare Borel bounded functions onN. Next, we remark that for any bounded, Borel functiongonN andu≥0, the functionPu−sg(Xs(ω)) on [0, u]×Ω isβ[0, u]× Fu^◦ measurable. Indeed, using the monotone class the- orem, we may consider just the case wheregis the trace onN of a continuous function on F, and using the right continuity of the semigroup (P_t) we have the approximation

Pu−sg(Xs(ω)) = lim

n hⁿ(s, ω), wherehn(s, ω) =P_u−ku

2ng(Xs(ω)) for ^ku_2n ≤s < ^(k+1)_2n ^u.

The measurability claimed above being now clear, we return to (2.6).

For each k= 0, . . . , n−1, the function 1_[tk,tk+1)(s)(Y_s−Y_s+1

n)(ω)f₁◦X_t1(ω). . . f_k◦X_tk(ω)P_tk+1−sg_k(X_s(ω)) isβ[0,∞)× F-(in factβ[0,∞)× F_tk+1+¹

n-) measurable.

Taking their sum, by Fubini theorem and the well known fact that the mapping x → E^xH is universally measurable for any F-measurable (bounded or positive) random variable H, the universal measurability of the mappings x→E^x[ϕB_tⁿ] considered above is now clear. Finally, it remains to make a standard regularization of the “process” (B_t) by considering

a¹_r = sup

srational s≤r

B_s for rational r≥0, and

a²_t =a¹_t+(right limit along the rationals > t), A_t=a²_t for any t≥0 if a²₀ = 0,A_t= 0 for any t≥0 if a²₀ = 0.

The process (At) is r.c., (Ft)-adapted (we used the fact that (Ft) is right continuous) andP^x-indistinguable from (B_t^x) for anyx∈N, which completes the proof.

Remarks. Our result implies the first half of the well known result con- cerning the representation of finitep-potentials of class (D) byp-additive func- tionals (see [1, XV, 2]). It points out that thep-additivity of the process (At) is exclusively a consequence of thep-homogeneity of the process (Y_t), namely that Yt = e^−ptf ◦Xt, where f is a p-excessive finite function on F. Another application of our result is the extension of [1, XV, 7b)] to the case of Ray semigroups, by the possibility to represent supermartingales Yt = f ◦X_t^T,

wheref is a finite excessive function and T is a stopping time relative to (Ft) such that (Y_t) is a supermartingale of class (D) (relative toP^xfor anyx∈N).

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Hermann, Paris, 1975, 1980, 1986.

[2] R.K. Getoor, Markov Processes: Ray Processes and Right Processes. Lecture Notes in Math.440. Springer, Berlin, 1975.

[3] V. Grecea,Positive bilinear mappings associated with stochastic processes.In:S´eminaire de Probabilit´esXXXVIII. Lecture Notes in Math.1857, pp. 145–157. Springer, Berlin, 2005.

[4] C.G. Herz,Bounded mean oscillation and regulated martingales.Trans. Amer. Math. Soc.

19(1974), 199–215.

[5] D. Lepingle,Sur la repr´esentation des sauts des martingales.In:S´eminaire de Probabilit´es XI, Lecture Notes in Math.581, pp. 418–434. Springer, Berlin, 1977.

[6] D. Lepingle,Sur certains commutateurs de la th´eorie des martingales.In: S´eminaire de Probabilit´esXII, Lecture Notes in Math.649, pp. 238–147. Springer, Berlin, 1978.

Received 20 October 2006 Romanian Academy

“Simion Stoilow” Institute of Mathematics P.O. Box 1-764, 014700 Bucharest, Romania

valentin.grecea@imar.ro